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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2015 Canada National Olympiad
2015 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(4)
5
1
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prime number dividing $a^N+b^n+c^N$ !CMO 2015 P5
Let
p
p
p
be a prime number for which
p
−
1
2
\frac{p-1}{2}
2
p
−
1
is also prime, and let
a
,
b
,
c
a,b,c
a
,
b
,
c
be integers not divisible by
p
p
p
. Prove that there are at most
1
+
2
p
1+\sqrt {2p}
1
+
2
p
positive integers
n
n
n
such that
n
<
p
n<p
n
<
p
and
p
p
p
divides
a
n
+
b
n
+
c
n
a^n+b^n+c^n
a
n
+
b
n
+
c
n
.
4
1
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circumcircle and altitudes! CMO 2015 P4
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with circumcenter
O
O
O
. Let
I
I
I
be a circle with center on the altitude from
A
A
A
in
A
B
C
ABC
A
BC
, passing through vertex
A
A
A
and points
P
P
P
and
Q
Q
Q
on sides
A
B
AB
A
B
and
A
C
AC
A
C
. Assume that
B
P
⋅
C
Q
=
A
P
⋅
A
Q
.
BP\cdot CQ = AP\cdot AQ.
BP
⋅
CQ
=
A
P
⋅
A
Q
.
Prove that
I
I
I
is tangent to the circumcircle of triangle
B
O
C
BOC
BOC
.
3
1
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combinatorial game!!! CMO 2015 P3
On a
(
4
n
+
2
)
×
(
4
n
+
2
)
(4n + 2)\times (4n + 2)
(
4
n
+
2
)
×
(
4
n
+
2
)
square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of
n
n
n
, what is the largest positive integer
k
k
k
such that there must be a row or column that the turtle has entered at least
k
k
k
distinct times?
2
1
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geometric inequality with altitudes! CMO 2015 P2
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with altitudes
A
D
,
B
E
,
AD,BE,
A
D
,
BE
,
and
C
F
CF
CF
. Let
H
H
H
be the orthocentre, that is, the point where the altitudes meet. Prove that
A
B
⋅
A
C
+
B
C
⋅
C
A
+
C
A
⋅
C
B
A
H
⋅
A
D
+
B
H
⋅
B
E
+
C
H
⋅
C
F
≤
2.
\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.
A
H
⋅
A
D
+
B
H
⋅
BE
+
C
H
⋅
CF
A
B
⋅
A
C
+
BC
⋅
C
A
+
C
A
⋅
CB
≤
2.